Root mean squar distance

A polymer chain can be approximated by a set of balls connected by springs. The springs account for the elastic behaviour of the chain and the beads are subject to viscous forces. In the Rouse model [35], the elastic force due to a spring connecting two beads is f= bAr, where Ar is the extension of the spring and the spring constant is ii = rtRis the root-mean-square distance of two successive beads. The viscous force that acts on a bead is... 

A molecular fitting algorithm requires a numerical measure of the difference between two structures when they are positioned in space. The objective of the fitting procedure is to find the relative orientations of the molecules in which this function is minimised. The most common measure of the fit between two structures is the root mean square distance between pairs of atoms, or RMSD ... 

A similarity measure is required for quantitative comparison of one strucmre with another, and as such it must be defined before the analysis can commence. Structural similarity is often measured by a root-mean-square distance (RMSD) between two conformations. In Cartesian coordinates the RMS distance dy between confonnation i and conformation j of a given molecule is defined as the minimum of the functional... 

Root-mean-square end-to-end distance, which effectively takes account of the average distance between the first and the last segment in the macromolecule, and is always less that the so-called contour length of the polymer. This latter is the actual distance from the beginning to the end of the macromolecule travelling along the covalent bonds of the molecule s backbone. Radius of gyration, which is the root-mean-square distance of the ele-... 

The root-mean-square distance Vr separating the ends of the polymer chain is a convenient measure of its linear dimensions. The dissymmetry coefficient will be unity for (VrV 0< l and will increase as this ratio increases. 

The dimension which enters into the actual calculation is the root-mean-square distance / (or radius of gyration ) of an element from the center of gravity. However, it may be shown that s —r /Q, which has been substituted to replace with /P in the expression leading to Eq. (32). 

The evaluation of the elastic free energy AFei rests on the assumption that the root-mean-square distance between the ends of the chain is distorted by the same factor a representing the linear expansion of the spatial distribution. As in the treatment of the swelling of network... 

This remarkably simple treatment suffers one serious deficiency the value of remains quantitatively undefined. More or less intuitively it has been suggested by various investigators that should increase as the root-mean-square end-to-end distance /for a linear chain, or, more generally, as the root-mean-square distance /s2 q beads from the center of any polymer molecule, linear or branched. Accepting this postulate unquestioningly, we should then have/o proportional to and [rj] proportional to These conclusions happen to... 

The radius, R, of each electron cloud is taken to be the root mean square distance of the electron from the center of the cloud. The equation of a sphere of radius, r, is x2 + y2 + z2 = r2, so on average ... 

For random coils, is directly proportional to the contour length. If n is the number of main chain atoms in the chain, = an. The parameter a is relatively insensitive to environment (21), and has been calculated for a number of polymers from strictly intramolecular considerations using the rotational isomeric model (22). The root-mean-square distance of segments from the center of gravity of the coil is called the radius of gyration S. The quantity S3 is an approximate measure of the pervaded volume of the coil. For Gaussian coils,... 

The root-mean-square distance (z2)122 of segments from the interface is calculated from... 

Fig. 3. Root mean-square distance of segments relative to pf1 from the interface as a function of molecular weight421 n has the same meaning as in Fig. 2...

In the pulmonary region, air velocities are too low to impact particles small enough to reach that region, and the mechanisms of deposition are sedimentation and Brownian diffusion. The efficiency of both processes depends on the length of the respiratory cycle, which determines the stay time in the lung. If the cycle is 15 breaths/min, the stay time is of the order of a second. Table 7.1 shows the distance fallen in one second and the root mean square distance travelled by Brownian diffusion in one second by unit density particles (Fuchs, 1964). Sedimentation velocity is proportional to particle density, but Brownian motion is independent of density. Table 7.1 shows that sedimentation of unit density particles is more effective in causing deposition than Brownian diffusion when dp exceeds 1 pm, whereas the reverse is true if dp is less than 0.5 pm. For this reason, it is appropriate to use the aerodynamic diameter dA equal to pj dp when this exceeds 1 pm, but the actual diameter for submicrometre particles. 

Under these assumptions, the probability of finding a carbon dioxide molecule MX from its starting point after N collisions is mathematically exactly the same as the coin toss probability of M more heads than tails. The root-mean squared distance traveled... 

The conformational behavior of Aim inserted in a POPC bilayer has been investigated using the same system as described above [87]. To insert the Aim molecule, a hole was introduced by restraints and one POPC molecule was removed. The Aim was placed into the bilayer, the system was hydrated, and one sodium ion added. A second simulation was performed in which no sodium ion was added and Glu18 was protonated instead. During the simulation period of 1000 ps, the Aim molecule remained in its a-helical conformation, showing only small fluctuations in the root mean square distance (RMSD) of the Ca atoms underlining its conformational stabilization by the lipid bilayer. 

Root mean square distance from center of each cluster... 

The rapid diffusibUity of NO has critically important imphcations for its chemistry in the biological setting. The speed with which NO moves by random diffusion can be illustrated by consideration of its root mean square distance of displacement, which describes the distance a single NO molecule will move in any time interval based on its diffusion constant D (which is similar for aqueous solution and also tissue (brain) ) ... 

A radius of gyration in general is the distance from the center of mass of a body at which the whole mass could be concentrated without changing its moment of rotational inertia about an axis through the center of mass. For a polymer chain, this is also the root-mean-square distance of the segments of the molecule from its center of mass. The radius of gyration is one measure of the size of the random coil shape which many synthetic polymers adopt in solution or in the amorphous bulk state. (The radius of gyration and other measures of macromolecular size and shape are considered in more detail in Chapter 4.)... 

The root-mean-square distance will be assumed as much larger than the distances producing nonnegligible values of (r), so that the exponential of Wf, t) may be expanded to the term. The function e(r) is positive (i.e., repulsive) and negative (attractive) at smaller and larger distances, respectively, and we shall separately consider the integrals Je(r)c( r and J r e(r)d r over these two distance ranges. We get... 

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